1. Chapter 2 – Polynomial and Rational Functions 2.3 – Real Zeros of Polynomial Functions

2. Long Division Ex: What is 27419 / 19? 1927419 01443 19 84 76 81 76 59 57 2

3. Long Division Ex: What is (6x 3 -19x 2 +16x – 4) / (x – 2)? x – 26x 3 – 19x 2 + 16x – 4 6x 2 – 7x+ 2 6x 3 – 12x 2 -7x 2 -7x 2 + 14x 2x 2x – 4 0 + 16x – 4 No remainder! Hooray!

4. Long Division Ex: What is (x 2 + 3x + 5) / (x + 1)? x + 1x 2 + 3x + 5 x+ 2 x 2 + x 2x 2x + 2 3 Remainder! Put this over the divisor! + 5

5. Long Division Ex: What is (4x 3 - 8x 2 +6) / (2x – 1)? 2x - 14x 3 – 8x 2 + 0x + 6 2x 2 – 3x- 3/2 4x 3 – 2x 2 -6x 2 -6x 2 + 3x -3x -3x +3/2 9/2 + 0x + 6

6. 1. None 2. 3. 4. 5. Divide 4x 2 – 6x + 7 by x + 3. What is the remainder?

7. 1. 2. 3. 4. Divide x 3 – 1 by x – 1. ***Put a zero for the coefficient of missing terms!!!

8. Synthetic Division When dividing binomials of the form x – k, use a shortcut called synthetic division! To divide ax 3 + bx 2 + cx + d by x – k… k a b c d a ka r Coefficients of quotient (answer) Coefficients of dividend Remainder Vertically – add terms Diagonally – multiply terms

9. Synthetic Division When dividing binomials of the form x – k, use a shortcut called synthetic division! Ex: Divide x 3 – 2x 2 + 5x – 3 by x – 3. ◦Set up the coefficients first… ◦…then multiply diagonally and add vertically! ◦So the quotient is x 2 + x + 8, and the 21 becomes the remainder. 3 1 -2 5 -3 1 3 211 3 8 24

10. Synthetic Division Ex: Divide x 5 – 2x 3 + 10x – 8 by x + 1. ◦Remember to put zeros for missing exponent terms! -1 1 0 -2 0 10 -8 1 1 1 1 9 -9 -17

11. Divide 2x 3 – 6x 2 + 7x – 4 by x – 2. What is the remainder? 1. 2. 3. 4. 5. None

12. Rational Zero Test Every rational zero of a polynomial f will be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. Ex: Find the possible rational zeros of f(x) = 2x 3 + 3x 2 – 8x + 3. ◦Possible rational zeros:

13. Factoring w/synthetic division If synthetic division gives a remainder of zero, then the divisor is a factor of the polynomial! Ex: Factor f(x) = 2x 3 + 3x 2 – 8x + 3. ◦On the previous slide, we determined the rational zeros, and of those rational zeros, only -1, 1, -3, and 3 were integers. ◦Guess and check using synthetic division! -1 2 3 -8 3 2 -2 121 -9 9 -1 is not a zero! 1 2 3 -8 3 2 2 05 5 -3 1 is a zero!

14. Factoring w/synthetic division Ex: Factor f(x) = 2x 3 + 3x 2 – 8x + 3. (cont.) ◦What’s left? ◦2x 2 + 5x – 3 ◦Because it’s a quadratic, you can factor this using either normal factoring or synthetic division. ◦2x 2 + 5x – 3  (2x – 1)(x + 3) ◦So the final factorization is (x – 1)(2x – 1)(x + 3).

15. Ex: Find all zeros of f(x) = 3x 3 + 4x 2 – 17x – 6. ◦Guess and check using synthetic division! ◦Our possible zeros are ±1, ±2, ±3, ±6, and ±1/3 ◦Answer: x = 2, -3, and -1/3 Ex: Find all zeros of f(x) = x 4 + x 3 - 4x 2 - 2x + 4. ◦Our possible rational zeros are ±1, ±2, ±4 ◦This factors to (x 2 – 2)(x + 2)(x – 1) ◦The (x 2 – 2) doesn’t factor, so use quadratic formula to find the REAL zeros of ◦Answer: Ex: Find all real zeros of f(x) = x 3 - x 2 - 4x - 2

16. Algebraically, find the zeros of f(x) = 3x 3 + x 2 – 8x + 4. 1. -2, -1, 2/3 2. 1, -2, 1.5 3. 1, 2/3, -2 4. 1.5, 2, -1